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\author{王立庆（2022级数学与应用数学1班）}
\title{高等代数(一)教案}

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\begin{document}

\maketitle

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\section{(3.1)线性方程组和行列式}

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%\subsection{内容提要}
%\begin{itemize}
%\item  线性方程组的三种形式
%\item  二阶和三阶行列式的定义，克拉默公式
%\end{itemize}

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\subsection{学习目标}
\begin{itemize}
\item  理解线性方程组的三种形式。
\item  计算二阶和三阶行列式的值，应用克拉默公式解方程组。
\end{itemize}

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\subsection{相关问题}
\begin{enumerate}
\item  写出线性方程组的一般形式。
\item  用三种形式，写出两个未知数两个线性方程，组成的线性方程组。
\item  写出二阶行列式的定义，并导出二元线性方程组的求解公式。
\item  写出三阶行列式的定义，并导出三元线性方程组的求解公式。
\item  解释平行四边形的面积为什么可以写成一个二阶行列式。
\item  解释平行六面体的体积为什么可以写成一个三阶行列式。
\item  说出矩阵与行列式的几点区别与联系。

\end{enumerate}

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\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 4em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{3-1}% root
child{ node {克拉默公式} 
	child{ node {由二阶到三阶} }
	child{ node {由一阶到二阶} }
	}
child{ node {行列式的几何意义} 
	child{ node {三阶} }
	child{ node {二阶} }
	}
child{ node {行列式的计算} 
	child{ node {三阶} }
	child{ node {二阶} }
	}
child{ node {线性方程组的表现形式} 
	child{ node {向量形式} }
	child{ node {矩阵形式} }
	child{ node {中学形式} }
}; 


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\section{(3.2)排列}

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%\subsection{内容提要}
%\begin{itemize}
%\item  
%\item  
%\end{itemize}

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\subsection{学习目标}
\begin{itemize}
\item  理解排列、对换、奇偶排列的概念。
\item  计算一个排列的反序数。
\item  证明对排列的一次对换正好改变这个排列的奇偶性。
\end{itemize}

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\subsection{相关问题}
\begin{enumerate}
\item  $n$个数字$1,2,\cdots,n$ 的一个排列，指的是什么？写出数字 $1,2,3$ 的所有排列。
\item  什么是排列的一个反序？什么是排列的反序数？计算排列 34251 的反序数。
\item  什么是奇排列，什么是偶排列？判断排列 345216 是奇排列还是偶排列。
\item  什么是对换？将排列 34521通过一系列的对换，变成排列 12345. 
\item  证明：数字 $1,2,\cdots,n$ 的任意两个排列，可以通过一系列的对换，来相互转换。
\item  证明：每次对换，正好改变排列的奇偶性。
\item  计算 6个数码 $1,2,3,4,5,6$ 的排列，分别有多少个奇排列和偶排列。

\end{enumerate}

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\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 4em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{3-2}% root
child{ node {对换} 
	child{ node {对换对排列的影响} }
	child{ node {概念} }
	}
child{ node {排列} 
	child{ node {偶排列} }
	child{ node {奇排列} }
	child{ node {反序数的概念} }
}; 


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\section{(3.3) $n$ 阶行列式}

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%\subsection{内容提要}
%\begin{itemize}
%\item  
%\item  
%\end{itemize}

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\subsection{学习目标}
\begin{itemize}
\item  掌握一般行列式的定义，理解每一项前的正负号的规律。
\item  从定义出发理解行列式的基本性质。
%\item  
\end{itemize}

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\subsection{相关问题}
\begin{enumerate}
\item  写出三阶行列式的计算表达式。找出指标的规律。
\item  用求和符号来重写三阶行列式的计算表达式。使用反序数的符号 $\pi(j_1j_2j_3)$.
\item  写出 $n$ 阶行列式的定义。
\item  按照定义计算一个四阶行列式的值。
\item  在四阶行列式 $D=\vert a_{ij}\vert$ 中，用两种方式，计算 $a_{24}a_{33}a_{12}a_{41}$ 前面的正负号。
\item  证明：行列式和它的转置行列式，有相同的取值。
\item  证明行列式交换两行，其值变成相反数。
\item  证明：如果行列式有两行相同，则其值为零。
\item  证明：行列式的某一行若有公因子则可以提出来。
\item  证明：如果行列式的某一行可以写成两行的和，那么这个行列式可以写成两个行列式的和。这两个行列式的其余行与原行列式相同，就这一行拆成了两行。
\item  把行列式的某一行乘以一个数，加到另一行，这个行列式的值不发生改变。
\item  两个计算的例子。
%\item  


\end{enumerate}

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\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 5em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{3-3}% root
child{ node {行列式的基本性质} 
	child{ node {第三类}}%某一行乘以同一个数再加到另一行} }
	child{ node {第二类}}%某一行有公因子} }
	child{ node {第一类}}%交换两行} }
	child{ node {分拆} }	
	child{ node {转置} }
	}
child{ node {行列式的定义} 
	child{ node {$n$阶} }
	child{ node {四阶} }
	child{ node {三阶} }
}; 


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\section{(3.4)子式和代数余子式、按行按列展开}

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%\subsection{内容提要}
%\begin{itemize}
%\item  
%\item  
%\end{itemize}

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\subsection{学习目标}
\begin{itemize}
\item  理解子式、余子式、代数余子式的概念。
\item  掌握行列式按行或按列展开的计算方法。
%\item  
\end{itemize}

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\subsection{相关问题}
\begin{enumerate}
\item  什么是一个 $k$ 阶子式？例子：找出四阶行列式的所有2阶子式。
\item  什么是行列式 $D=\vert a_{ij}\vert_{n\times n}$ 的元素 $a_{ij}$ 的余子式？什么是元素 $a_{ij}$ 的代数余子式？
\item  证明：若行列式 $D=\vert a_{ij}\vert_{n\times n}$ 的第 $i$ 行只有 $a_{ij}$ 不等于零，则 
\[ D= a_{ij}A_{ij}, \]
这里 $A_{ij}$ 是 $a_{ij}$ 的代数余子式。
\item  行列式可以按行展开。举例说明这种算法。
\item  证明：行列式的某一行的元素，与另一行的元素的代数余子式，两两相乘相加，结果总是零。
\item  例子：计算行列式的值。
\item  拉普拉斯展开是一种什么样的展开？
\end{enumerate}

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\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 4em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{3-4}% root
child{ node {计算} 
	child{ node {Laplace 展开} }
	child{ node {按列展开} }
	child{ node {按行展开} }
	}
child{ node {概念} 
	child{ node {代数余子式} }
	child{ node {余子式} }
	child{ node {子式} }
}; 


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\section{(3.5)克拉默规则}

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%\subsection{内容提要}
%\begin{itemize}
%\item  
%\item  
%\end{itemize}

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\subsection{学习目标}
\begin{itemize}
\item  熟练计算代数余子式。
\item  掌握运用克拉默规则。
\item  理解克拉默规则的证明。
\end{itemize}

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\subsection{相关问题}
\begin{enumerate}
\item  什么是克拉默规则？
\item  在 $n=2$ 与 $n=3$ 的情形，写出克拉默规则的证明。
\item  用克拉默规则，求解线性方程组
\[\left\{\begin{array}{rcl}
2x_1 +x_2 -5x_3 +x_4 &=& 8 \\
x_1 -3x_2 +0x_3 -6x_4 &=& 9 \\
0x_1 +2x_2 -x_3+2x_4 &=& -5 \\
x_1 +4x_2 -7x_3 +6x_4 &=& 0 \\
\end{array}\right.\]
\item  如何在未知数个数与方程个数不相等的时候，使用克拉默公式？
\item  代数余子式 $A_{ij}$ 有巧妙的应用。举例说明，使用代数余子式，来同时消去未知数 $x_2,x_3$, 
\[\left\{\begin{array}{rcl}
x_1 +2x_2 +3x_3 &=& 4 \\
5x_1 +6x_2 +7x_3 &=& 8 \\
9x_1 +2x_2 +5x_3 &=& 7 \\
\end{array}\right.\]
%\item  
%\item  
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 4em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{3-5}% root
child{ node {代数余子式} 
	child{ node {运用} }
	child{ node {计算} }
	}
child{ node {克拉默规则} 
	child{ node {运用} }
	child{ node {证明} }
	child{ node {内容} }
}; 

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\section{(4.1)消元法}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{内容提要}
%\begin{itemize}
%\item  
%\item  
%\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{学习目标}
\begin{itemize}
\item  理解线性方程组的初等变换的概念。
\item  理解线性方程组的系数矩阵和增广矩阵的概念。
\item  熟练掌握使用行初等变换将矩阵化为行最简形。
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{相关问题}
\begin{enumerate}
\item  用消元法求解线性方程组
\[\left\{\begin{array}{rcl}
(1/2)x_1 +(1/3)x_2 +x_3 &=& 1 \\
x_1 +(5/3)x_2 +3x_3 &=& 3 \\
2x_1 +(4/3)x_2 +5x_3 &=& 2 \\
\end{array}\right.\]
\item  什么是线性方程组的初等变换？证明线性方程组在初等变换前后的解集是相同的。
\item  什么是矩阵？什么是线性方程组的系数矩阵和增广矩阵？
\item  什么是矩阵的行初等变换和列初等变换？
\item  对任意一个 $m\times n$ 矩阵，总能通过行初等变换和第一种列初等变换，化为下述形式的矩阵，
\[\begin{pmatrix} E_r  & * \\ O & O \end{pmatrix}. \]
\item  设 $A$ 是一个 $2\times 3$ 矩阵。按照定理4.1.2. 进行行初等变换和第一类初等变换，可以得到哪些不同类型的结果？
\item  什么是行最简形？对 $3\times 4$ 矩阵，可以得到哪些不同类型的行最简形？
\item  几个例子。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 4em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{4-1}% root
child{ node {矩阵} 
	child{ node {行最简形} }
	child{ node {三类列变换} }
	child{ node {三类行变换} }
	}
child{ node {线性方程组} 
	child{ node {增广矩阵} }
	child{ node {系数矩阵} }
	child{ node {三类初等变换} }
}; 


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\section{(4.2)矩阵的秩、线性方程组可解的判别法}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{内容提要}
%\begin{itemize}
%\item  
%\item  
%\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{学习目标}
\begin{itemize}
\item  理解子式的概念，理解矩阵的秩的概念。用初等变换计算矩阵的秩。
\item  理解线性方程组有解的充分必要条件。
\item  理解线性方程组无解、有唯一解、有无穷多解的判别条件。
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{相关问题}
\begin{enumerate}
\item  什么是一个矩阵的 $k$ 阶子式？
\item  计算下述矩阵的所有$k$ 阶子式，其中 $k=1,2,3$,
\[ A=\begin{pmatrix} 1&2&3&4 \\ 5&6&7&8 \\ 9&0&1&2 \end{pmatrix}. \]
\item  什么是矩阵的秩？计算上述矩阵的秩。
\item  证明：初等变换不改变矩阵的秩。
\item  例子：用初等变换计算矩阵的秩。
\item  证明：线性方程组有解的充分必要条件是它的系数矩阵和增广矩阵有相同的秩。
\item  证明：设线性方程组的系数矩阵和增广矩阵的秩相等而且等于 $r$. 设未知数的个数是 $n$. 则当 $r=n$ 时，该线性方程组有唯一解。当 $r<n$ 时，该线性方程组有无穷多解。
\item  例子：讨论线性方程组的解的情况。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 4em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{4-2}% root
child{ node {线性方程组} 
	child{ node {有解} 
            	child{ node {有无穷多解} }
            	child{ node {有唯一解} }
            	}	
	child{ node {无解} }
	}
child{ node {矩阵的秩} 
	child{ node {计算} }
	child{ node {行列式秩} }
	child{ node {子式} }
}; 


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\section{(4.3)线性方程组的公式解}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{内容提要}
%\begin{itemize}
%\item  
%\item  
%\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{学习目标}
\begin{itemize}
\item  使用矩阵的秩，去掉线性方程组中多余的方程。
\item  理解非齐次线性方程组的导出组的概念。
\item  理解齐次线性方程组有非零解的充分必要条件。
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{相关问题}
\begin{enumerate}
\item  简化下述线性方程组，但不产生新的系数和常数项，
\[\left\{\begin{array}{rcl}
x_1 +2x_2 -x_3 &=& 2 \\
2x_1 -3x_2 +x_3 &=& 3 \\
4x_1 +x_2 -x_3 &=& 7 \\
\end{array}\right.\]
\item  证明：设线性方程组的系数矩阵和增广矩阵的秩都是 $r$. 那么一定可以从这个线性方程组里选出 $r$ 个方程，由这 $r$ 个方程形成的线性方程组与原线性方程组具有相同的解集。
\item  设下述线性方程组的系数矩阵和增广矩阵的秩都等于2，且 $a_1b_3 - a_3b_1 \neq 0$. 则第三个方程一定可以从前两个方程通过组合得到。
\[\left\{\begin{array}{rcl}
a_1x_1 +a_2x_2 +a_3x_3 &=& u \\
b_1x_1 +b_2x_2 +b_3x_3 &=& v \\
c_1x_1 +c_2x_2 +c_3x_3 &=& w \\
\end{array}\right.\]

\item  对一个线性方程组，用公式写出它的解的步骤是什么？
\item  什么是齐次的线性方程组？什么是非齐次的线性方程组？什么是非齐次线性方程组的导出组？
\item  齐次线性方程组有非零解的充分必要条件是系数矩阵的秩小于未知数的个数。
\item  例子。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 4em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{4-3}% root
child{ node {线性方程组的分类} 
	child{ node {非齐次} 
		child{ node {导出组} }
	}
	child{ node {齐次} }
	}
child{ node {线性方程组的公式解} 
	child{ node {克拉默公式} }
	child{ node {略去多余的方程} }
	child{ node {矩阵的秩} }
}; 


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\newpage
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\section{(5.1)矩阵的运算}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{内容提要}
%\begin{itemize}
%\item  
%\item  
%\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{学习目标}
\begin{itemize}
\item  掌握矩阵的加法、数乘、乘法、转置运算。
\item  理解零矩阵、单位矩阵、负矩阵的概念。
\item  理解矩阵的运算规律。
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{相关问题}
\begin{enumerate}
\item  什么时候两个矩阵称为是相等的？
\item  数与矩阵的乘法是怎样进行的？两个矩阵的加法是怎样进行的？
\item  什么是零矩阵？有多少个零矩阵？什么是负矩阵？矩阵的减法是怎么定义的？
\item  什么时候两个矩阵可以相乘？矩阵的乘法是怎么进行的？
\item  证明：矩阵的乘法满足结合律。即 $(AB)C=A(BC)$. 
\item  什么是单位矩阵？有多少个单位矩阵？
\item  证明：单位矩阵如果可以与别的矩阵相乘，那么结果就是那个矩阵。
\item  一个方阵的幂次是怎么计算的？
\item  设 $A$ 是一个方阵，设 $f(x)$ 是一个多项式，则 $f(A)$ 是怎么计算的？
\item  矩阵的转置是怎么进行的？矩阵的转置，与加法、数乘、乘法一起，有哪些运算规律？
\item  例子。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 4em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{5-1}% root
child{ node {运算规律} 
	child{ node {分配律} }
	child{ node {交换律} }
	child{ node {结合律} }
	}
child{ node {特殊矩阵} 
	child{ node {单位矩阵} }
	child{ node {零矩阵} }
	}
child{ node {矩阵的运算} 
	child{ node {幂次} }
	child{ node {转置} }
	child{ node {乘法} }
	child{ node {数乘} }
	child{ node {加法} }
}; 


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\section{(5.2)可逆矩阵、矩阵乘积的行列式}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{内容提要}
%\begin{itemize}
%\item  
%\item  
%\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{学习目标}
\begin{itemize}
\item  理解逆阵的概念，使用初等变换和伴随矩阵两种方法计算逆阵。
\item  理解初等矩阵的概念，理解左乘初等矩阵与初等行变换之间的联系。
\item  理解初等变换对矩阵的秩的影响。
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{相关问题}
\begin{enumerate}
\item  什么时候称一个矩阵是可逆的？
\item  什么是初等矩阵？解释左乘初等矩阵与行初等变换的关系，右乘初等矩阵与列初等变换的关系。
\item  证明：一个 $m\times n$ 矩阵总可以通过若干次初等变换，化成如下形式的矩阵，
\[ \begin{pmatrix} E_r&O \\ O&O \end{pmatrix}, \]
其中 $E_r$ 是 $r$ 阶的单位阵，三个 $O$ 是阶数可能不同的三个零矩阵。
\item  证明：一个矩阵是可逆的，当且仅当这个矩阵可以写成一些初等矩阵的乘积。
\item  证明：一个 $n$ 阶矩阵可逆，当且仅当这个矩阵的秩等于 $n$. 
\item  证明：一个矩阵可逆，当且仅当这个矩阵的行列式的值不等于零。
\item  什么是伴随矩阵？
\item  证明伴随矩阵的基本公式
\[ AA^* = \det(A)E_n\]
\item  使用伴随矩阵来推导克拉默公式。
\item  证明：对任意两个 $n$ 阶矩阵，都有
\[\det(AB) = \det(A)\det(B). \]
\item 证明：一个矩阵乘以一个可逆矩阵，不改变这个矩阵的秩。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 4em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{5-2}% root
child{ node {矩阵的秩} 
	child{ node {初等变换对秩的影响} }
	child{ node {可逆矩阵的秩} }
	}
child{ node {初等矩阵} 
	child{ node {与列初等变换的联系} }
	child{ node {与行初等变换的联系} 
		child{ node {行列式乘积公式} }
		}
	child{ node {定义} }
	}
child{ node {逆阵} 
	child{ node {计算} 
        		child{ node {伴随矩阵求逆阵} }
        		child{ node {初等变换求逆阵} }
		}
	child{ node {定义} }
}; 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%\setcounter{section}{0}

\section{(5.3)矩阵的分块}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{内容提要}
%\begin{itemize}
%\item  
%\item  
%\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{学习目标}
\begin{itemize}
\item  理解矩阵分块的方法，理解分块矩阵的加法、数乘和乘法运算。
\item  计算分块矩阵的行列式的值和逆矩阵。
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{相关问题}
\begin{enumerate}
\item  解释分块矩阵的加法和数乘的运算规律。
\item  两个分块矩阵是如何相乘的？
\item  用先分块的方法，计算矩阵的乘积，
\begin{eqnarray*}
\begin{pmatrix} 1&0&0&0 \\ 0&1&0&0 \\ -1&2&1&0 \\ 1&1&0&1 \end{pmatrix}
\cdot
\begin{pmatrix} 1&0&3&2 \\ -1&2&0&1 \\ 1&0&4&1 \\ -1&-1&2&0 \end{pmatrix}.
\end{eqnarray*}
\item  设矩阵 $A,B$ 可逆，求下述分块矩阵的逆阵，
\begin{eqnarray*}
\begin{pmatrix} A&C \\ O&B \end{pmatrix}.
\end{eqnarray*}
\item  设矩阵 $A,B,C$ 可逆，求下述分块矩阵的逆阵，
\begin{eqnarray*}
\begin{pmatrix} A&O&O \\ O&B&O \\ O&O&C \end{pmatrix}.
\end{eqnarray*}
\item  设矩阵 $A,B$ 可逆，求下述分块矩阵的伴随矩阵，
\begin{eqnarray*}
\begin{pmatrix} A&C \\ O&B \end{pmatrix}.
\end{eqnarray*}
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 4em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{5-3}% root
child{ node {分块初等变换} 
	child{ node {对应的矩阵} 
		child{ node {应用于矩阵的秩的证明} }	
	}
	child{ node {与初等变换的关系} }
	}
child{ node {分块矩阵的计算} 
	child{ node {求伴随矩阵} }
	child{ node {求行列式值} }
	child{ node {求逆阵} }
	}
child{ node {分块矩阵} 
	child{ node {好处} }
	child{ node {基本运算规则} }
	child{ node {含义} }
}; 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%\setcounter{section}{0}
\section{(6.1)向量空间的定义和例子}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\subsection{内容提要}
%\begin{itemize}
%\item 概念：向量空间、向量、零向量、负向量。
%\item 重点：理解向量空间的概念和一些例子。从公理出发证明向量空间的两个运算的基本性质。%验证一个集合在指定运算下是否成为一个向量空间。
%\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{学习目标}
\begin{itemize}
\item  理解向量空间的概念和一些例子。
\item  按向量空间的公理，在给定的向量空间中进行线性运算。
\item  从公理出发证明向量空间的两个运算的基本性质。
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{相关问题}
\begin{enumerate}
\item 定义：称集合 $V$ 是数域 $F$ 上的一个{\color{red}向量空间}，是指：
    \begin{itemize}
    \item 数域 $F$ 中的元素称为数，集合 $V$ 中的元素称为{\color{red}向量}。
    \item 两个运算：
       \begin{enumerate}
       \item 数乘：$F\times V\to V: (k,\alpha)\mapsto k\alpha$.
       \item 向量加法：$V\times V \to V: (\alpha,\beta)\mapsto \alpha+\beta$.
       \end{enumerate}
    \item 符合两组公理：
       \begin{enumerate}
       \item 向量的加法有交换律、结合律、存在{\color{red}零向量}、存在{\color{red}负向量}。
       \item 数乘有两个分配律、连乘、一乘。
       \end{enumerate}
    \end{itemize}

\item 向量空间的例子：
    \begin{enumerate}
    \item 例子1：平面空间、立体空间。
    \item 例子2：实系数 $m\times n$ 矩阵全体组成的集合。
    \item 例子3：复数域看做实数域上的向量空间。
    \item 例子4：数域 $F$ 看做自身上的向量空间。
    \item 例子5：实系数一元多项式全体组成的集合。
    \item 例子6：定义在某闭区间上的所有实值函数全体组成的集合。
    \item 例子7：收敛于零的实数序列全体组成的集合。
    \end{enumerate}

\item 命题6.1.1. 给定向量空间，证明其零向量是唯一的，任意向量的负向量是唯一确定的。
\item 命题6.1.2. 
    \begin{enumerate}
    \item 数字零乘以任意向量的结果是什么？
    \item 任意数字乘以零向量的结果是什么？
    \item 数字乘以负向量的结果是什么？
    \item 数乘得到零向量的原因是什么？
    \end{enumerate}
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 4em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{向量空间}% root
child{ node {向量空间的基本性质} 
	child{ node {命题6.1.2} }
	child{ node {命题6.1.1} }
	}
child{ node {向量空间的例子} 
	child{ node {函数空间} }
	child{ node {矩阵空间} }
	child{ node {行向量空间} }
	}
child{ node {向量空间的概念} 
	child{ node {运算公理} }
	child{ node {运算} 
		child{ node {数乘} }
		child{ node {加法} }	
	}
}; 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\section{(6.2)子空间}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\subsection{内容提要}
%\begin{itemize}
%\item 概念：子空间、交子空间、和子空间。
%\item 重点：
%\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{学习目标}
\begin{itemize}
\item  理解子空间的概念。
\item  验证一个向量空间的子集是否成为一个子空间。
\item  理解交子空间与和子空间的概念。
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{内容讲解}
\begin{enumerate}
\item {\color{red}子空间}的定义：
    \begin{enumerate}
    \item 子空间是向量空间的非空子集。
    \item 这个子集中的向量，在原向量空间的加法和数乘运算下的，其结果仍在这个子集中。
    \item 子空间是{\color{red}向量子空间}的简称。
    \end{enumerate}
\item 例子：
    \begin{enumerate}
    \item 例子1：向量空间是它自身的子空间。只有零向量的子集是所有向量空间的子空间。
    \item 例子2：平面中过原点的直线是平面的子空间。立体空间中过原点的直线或平面是立体空间的子空间。
    \item 例子3：$F^n$ 中的一个子空间的例子：最后一个分量为零的所有向量组成的子集。
    \item 例子4：次数不超过给定值的所有多项式和零多项式组成的子集。
    \item 例子5：某个闭区间上的所有可微函数组成的子集。
    \end{enumerate}

\item 定理6.2.1. 子空间也是一个向量空间。注意从子空间和向量空间的概念出发来证明。

\item 定理6.2.2.判别一个子集是不是子空间，只需验证这个子集的向量在线性运算下的封闭性。\\ 注：向量的加法和数乘的综合运算，称为线性运算。
\item {\color{red}交子空间}的概念：两个子空间的交集，仍是一个子空间。这需要证明。
\item {\color{red}和子空间}的概念：两个子空间的并集，再放上向量加法能得到的所有向量。设 $U$ 与 $W$ 是向量空间 $V$ 的两个子空间，则它们的和子空间定义为下述子集：
$$U+W=\{\alpha+\beta \mid \alpha\in U, \beta\in W\}. $$

\item 命题：和子空间是包含这两个子空间的最小的子空间。

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 4em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{子空间}% root
child{ node {运算} 
	child{ node {和子空间} }
	child{ node {交子空间} }
	}
child{ node {例子} 
	child{ node {函数空间} }
	child{ node {矩阵空间} }
	child{ node {行向量空间} }
	}
child{ node {定义} 
	child{ node {运算封闭} }
	child{ node {非空子集} }
}; 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\section{(6.3)向量的线性相关性}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\subsection{内容提要}
%\begin{itemize}
%\item 概念：线性组合、、极大线性无关组。
%\item 重点：
%\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{学习目标}
\begin{itemize}
\item  理解向量组线性相关、线性无关的概念，理解向量组相互等价的概念。
\item  计算一个向量组的极大线性无关组。
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{内容讲解}
\begin{enumerate}
\item 定义：向量 $\alpha_1,\cdots,\alpha_r$ 的{\color{red}线性组合}是什么？
\item 例子：向量空间 $\mathbb{R}^3$ 中，给定三个向量 $\alpha_1,\alpha_2,\alpha_3$, 计算 $2\alpha_1-\alpha_2+3\alpha_3$ 是什么。

\item 称向量组 $\{\alpha_1,\cdots,\alpha_r\}$ {\color{red}线性相关}或{\color{red}线性无关}，是指什么？
%\item 定义：
%    \begin{enumerate}
%    \item 称向量组 $\{\alpha_1,\cdots,\alpha_r\}$ {\color{red}线性相关}，是指存在不全为零的数 $k_1,\cdots,k_r$, 使得其线性组合等于零向量，即有：\(k_1\alpha_1 +\cdots +k_r\alpha_r ={\bf 0}\).
%    \item 称向量组 $\{\alpha_1,\cdots,\alpha_r\}$ {\color{red}线性无关}，是指什么？
%    \end{enumerate}

\item 例子：
    \begin{enumerate}
    \item 若一个向量组中有零向量，则这个向量组一定是线性相关的。
    \item 单独一个非零向量，组成的向量组，是线性无关的。
    \end{enumerate}

\item 例子1：设 $F$ 是任意一个数域。设
\begin{eqnarray*}
\alpha_1=(1,2,3), \alpha_2=(2,4,6), \alpha_3=(3,5,-4);  \\
\beta_1=(1,0,0),\beta_2=(1,1,0),\beta_3=(1,1,1).
\end{eqnarray*}
则向量组 $\{\alpha_1,\alpha_2,\alpha_3\}$ 是线性相关的，向量组 $\{\beta_1,\beta_2,\beta_3\}$ 是线性无关的。（用定义来证）

\item 例子2：判断下述 $F^3$ 中的向量组是否线性相关：
\begin{eqnarray*}
\alpha_1=(1,-2,3), \alpha_2=(2,1,0), \alpha_3=(1,-7,9).
\end{eqnarray*}

\item 例子3：向量空间 $F[x]$ 中，向量组 $\{1,x,x^2,\cdots,x^n\}$ 是线性无关的。（用定义来证）

%\item 命题1-4.

\item 证明定理6.3.1. 向量组线性相关，当且仅当其中一个向量可以写成其余向量的线性组合。

\item 定义：称{\color{red}两个向量组相互等价}，是指什么？

\item 例子4：证明向量组 $\alpha_1=(1,2,3),\alpha_2=(1,0,2)$ 与向量组 $\beta_1=(3,4,8), \beta_2=(2,2,5),\beta_3=(0,2,1)$ 相互等价。（用定义来证）

\item 定义：一个向量组的{\color{red}极大线性无关组}，是指什么？

\item 例子5：求 $\alpha_1=(1,0,0),\alpha_2=(0,1,0),\alpha_1=(1,1,0)$ 的极大无关组。

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 4em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{向量组}% root
child{ node {极大线性无关组} 
	child{ node {几何含义} }
	child{ node {计算} }
	child{ node {定义} }
	}
child{ node {相互等价} 
	child{ node {方程含义} }
	child{ node {几何含义} }
	child{ node {定义} }
	}
child{ node {线性相关} 
	child{ node {方程含义} }
	child{ node {几何含义} }
	child{ node {定义} }
	}
child{ node {线性无关} 
	child{ node {方程含义} }
	child{ node {几何含义} }
	child{ node {定义} }
}; 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\section{(6.4)基和维数}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\subsection{内容提要}
%\begin{itemize}
%\item 概念：生成元，向量空间的基，标准基，向量空间的维数，直和。
%\item 重点：
%\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{学习目标}
\begin{itemize}
\item  理解基的概念和性质，证明维数公式，理解直和的概念和性质。
\item  计算一个子空间的维数，计算一个子空间的余子空间。
%\item  
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{内容讲解}
\begin{enumerate}
\item 由一个向量组生成的子空间 $\mathcal{L}(\alpha_1,\cdots,\alpha_n)$ 指的是什么？
\item 例1：什么是向量空间的{\color{red}生成元}？求 $F^n$ 的一组生成元。
\item 例2：多项式空间 $F[x]$ 中次数不超过 $n$ 的所有多项式组成的子空间，求一组生成元。
\item 定理6.4.1. 设 $\{\alpha_1,\alpha_2,\cdots,\alpha_n\}$ 是向量空间 $V$ 的一个不全为零的向量组，设 $\{\alpha_{i_1},\alpha_{i_2},\cdots,\alpha_{i_k}\}$ 是它的一个极大线性无关组。则这两个向量组生成的子空间是一样的。
\item 定义1:向量空间 $V$ 的{\color{red}一个基}，是指什么？
    %这个向量空间的一个向量组 $\{\alpha_1,\alpha_2,\cdots,\alpha_n\}$, 满足两个条件：
    %\begin{enumerate}
    %\item 这个向量组本身线性无关；
    %\item 这个向量空间的每一个向量都可以由这个向量组线性表示。
    %\end{enumerate}
\item 例3: $F^n$ 的{\color{red}标准基}。
\item 例4: 平面和立体空间的基。
\item 例5: 矩阵空间的基。
\item 定义2: 一个向量空间的{\color{red}维数}是指其任意一个基所含有的向量的个数。
\item 概念：{\color{red}有限维向量空间}和{\color{red}无限维向量空间}。
\item 定理6.4.2. 向量空间的每个向量都可以由一个基所唯一地线性表示。
\item 定理6.4.3. $n$ 维向量空间中的多于 $n$ 个向量一定线性相关。
\item 定理6.4.4. 有限维向量空间中的线性无关组都能扩充成一个基。
\item 定理6.4.5. 维数公式：设 $W_1$ 和 $W_2$ 是向量空间 $V$ 的有限维子空间，则 $W_1+W_2$ 也是有限维的，而且有
$\dim (W_1+W_2) = \dim W_1 + \dim W_2 - \dim (W_1\cap W_2)$. 
\item  定义3: 向量空间 $V$ 的子空间 $W$ 的一个{\color{red}余子空间}是指什么？
%\item  定义3: 向量空间 $V$ 的子空间 $W$ 的一个{\color{red}余子空间}是指这样一个子空间 $W'$, 满足 $V=W+W'$ 与 $W\cap W'=\{\theta\}$. 这里 $\theta$ 是这个向量空间的零向量。
\item 概念：{\color{red}直和} $V=W\oplus W'$ 是指什么？
\item 定理6.4.6. 直和的性质。（可当作练习来证）
\item 定理6.4.7. 余子空间一定存在。（可当作练习来证）

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 4em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{向量组}% root
child{ node {例子} 
	child{ node {函数空间} }
	child{ node {矩阵空间} }
	child{ node {行向量空间} }
	}
child{ node {子空间的基} 
	child{ node {直和} }
	child{ node {余子空间} }
	child{ node {定义} }
	}
child{ node {向量空间的基} 
	child{ node {向量空间的维数} }
	child{ node {定义} }
}; 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\section{(6.5)坐标}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\subsection{内容提要}
%\begin{itemize}
%\item 概念：向量关于一个基的坐标，由一个基到另一个基的过渡矩阵。
%\item 重点：计算过渡矩阵，计算一个向量在不同的基下的坐标。
%\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{学习目标}
\begin{itemize}
\item  理解坐标的概念，理解过渡矩阵的概念。
\item  计算向量空间的两个基之间的过渡矩阵，和向量在一个基下的坐标。
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{内容讲解}
\begin{enumerate}
\item 设 $V$ 是数域 $F$ 上的一个向量空间，设 $\{\alpha_1,\alpha_2,\cdots,\alpha_n\}$ 是 $V$ 的一个基。设 $\xi$ 是 $V$ 中的一个向量。则向量 $\xi$ 关于基 $\{\alpha_1,\alpha_2,\cdots,\alpha_n\}$ 的{\color{red}坐标}是指什么？
\item 例子1： 立体空间的向量关于一个基的坐标。
\item 例子2：$F^n$ 关于标准基的坐标。
\item 定理6.5.1. 已知向量 $\xi,\eta$ 关于一个基的坐标，求 $\xi+\eta$ 和 $a\xi$ 关于这个基的坐标。 
\item 概念：由基 $\{\alpha_1,\alpha_2,\cdots,\alpha_n\}$ 到基 $\{\beta_1,\beta_2,\cdots,\beta_n\}$ 的{\color{red}过渡矩阵}，指的是什么？
\item 定理6.5.2. 已知向量 $\xi$ 关于一个基 $\{\alpha_1,\alpha_2,\cdots,\alpha_n\}$ 的坐标，已知由基 $\{\alpha_1,\alpha_2,\cdots,\alpha_n\}$ 到基 $\{\beta_1,\beta_2,\cdots,\beta_n\}$ 的过渡矩阵，求向量 $\xi$ 关于基 $\{\beta_1,\beta_2,\cdots,\beta_n\}$ 的坐标。
\item 例子3：平面上两个基之间的过渡矩阵，与平面绕原点旋转的坐标变换公式。
\item 定理6.5.3. 向量空间的两个基之间的过渡矩阵互为逆阵。
\item 例子4：证明 $\mathbb{R}^3$ 中的一个向量组构成一个基，并求另一个向量关于这个基的坐标。
\item 例子5：计算 $\mathbb{R}^3$ 中的两个基之间的过渡矩阵。
%\item 
%\item 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 4em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{向量空间}% root
child{ node {例子} 
	child{ node {一般向量空间} }
	child{ node {立体} }
	child{ node {平面} }
	}
child{ node {两个基} 
	child{ node {坐标的变化} }
	child{ node {过渡矩阵} }
	}
child{ node {一个基} 
	child{ node {向量的坐标} }
}; 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newpage
\section{(6.6)向量空间的同构}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\subsection{内容提要}
%\begin{itemize}
%\item 概念：向量空间之间的同构。
%\item 重点：同构的构造。
%\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{学习目标}
\begin{itemize}
\item  理解同构的概念。
\item  验证两个向量空间之间的映射是否为同构映射。
\item  构造两个向量空间之间的同构。
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{内容讲解}
\begin{enumerate}
\item 设 $V$ 是数域 $F$ 上的一个 $n$ 维向量空间，取定 $V$ 的一个基 $\{\alpha_1,\alpha_2,\cdots,\alpha_n\}$, 如何构造一个映射 $f: V\to F^n,$
使得 $f$ 保持了 $V$ 和 $F^n$ 各自的向量加法和数乘运算？

\item 定义1: 设 $V$ 和 $W$ 是数域 $F$ 上的两个向量空间，一个映射 $f:V\to W$ 什么时候称为是一个{\color{red}同构映射}？

\item 定理6.6.1. 数域 $F$ 上的任意一个 $n$ 维向量空间 $V$ 都与 $F^n$ 同构。

\item 定理6.6.2. 设 $V$ 和 $W$ 是数域 $F$ 上的两个向量空间，设 $f:V\to W$ 是一个同构映射。则
\begin{enumerate}
\item $f(\theta)$ 是什么？这里 $\theta$ 是零向量。
\item $f$ 保持向量空间的线性运算。
\item $f(-\alpha)$ 是什么？这里 $-\alpha$ 是 $\alpha$ 的负向量。
\item $f$ 的逆映射也是一个同构映射。
\end{enumerate}

\item 定理6.6.3. 数域 $F$ 上两个有限维向量空间同构的充分必要条件是它们的维数相同。

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 4em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{两个向量空间}% root
child{ node {构造同构} 
	child{ node {使用一个基} }
	child{ node {映射的定义} }
	}
child{ node {同构的定义} 
	child{ node {保持数乘运算} }
	child{ node {保持加法运算} }
	child{ node {双射} }
}; 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\section{(6.7)矩阵的秩、齐次线性方程组的解空间}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{内容提要}
%\begin{itemize}
%\item 概念：矩阵的行空间、列空间、矩阵的秩。基础解系。
%\item 重点：理解矩阵的行秩和列秩，计算基础解系，理解线性方程组的解集的结构。
%\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{学习目标}
\begin{itemize}
\item  理解矩阵的行秩和列秩的概念。
\item  计算齐次线性方程组的基础解系。
\item  理解非齐次线性方程组的解集的结构。

\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{内容讲解}
\begin{enumerate}
\item 矩阵的{\color{red}行空间}和{\color{red}列空间}指的是什么？
\item 引理6.7.1. 设 $A$ 是一个 $m\times n$ 矩阵。设 $P$ 和 $Q$ 分别是 $m$ 阶和 $n$ 阶可逆矩阵。则 $A$ 和 $PA$ 有相同的行空间，$A$ 和 $AQ$ 有相同的列空间。

\item 定理6.7.1. 一个矩阵的行空间的维数等于列空间的维数，等于这个矩阵的秩。
\item 定理6.7.2. 数域 $F$ 上的 $n$ 个未知数的齐次线性方程组的一切解向量组成了 $F^n$ 的一个子空间，称为解空间。如果系数矩阵的秩是 $r$, 那么这个{\color{red}解空间}的维数等于 $n-r$. 

\item 问题：什么是{\color{red}基础解系}？\\ 回答：齐次线性方程组的解空间的任意一个基，就称为这个齐次线性方程组的的一个基础解系。

\item 例子1: 求下述齐次线性方程组的一个基础解系：
\begin{eqnarray*}
\left\{\begin{array}{rcl}
x_1-x_2+5x_3-x_4 &=& 0, \\
x_1+x_2-2x_3+3x_4 &=& 0, \\
3x_1-x_2+8x_3+x_4 &=& 0, \\
x_1+3x_2-9x_3+7x_4 &=& 0. \\
\end{array}\right.
\end{eqnarray*}

\item 定理6.7.3. 非齐次线性方程组 $Ax=b$ 的解与齐次线性方程组 $Ax=0$ 的解之间有什么联系？

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{知识结构}
\usetikzlibrary{trees}
\tikz [font = \footnotesize, grow = right, 
level 1/.style={sibling distance = 4em},
level 2/.style={sibling distance = 1em}, level distance = 4cm ]
\node{6-7}% root
child{ node {矩阵的秩} 
	child{ node {矩阵的子式} 
		child{ node {行列式秩} }	
	}
	child{ node {矩阵的列空间} 
		child{ node {列秩} }	
	}
	child{ node {矩阵的行空间}
		child{ node {行秩} }
		}
	}
child{ node {非齐次线性方程组} 
	child{ node {解集的结构} }
	child{ node {有解的判别条件} }
	child{ node {导出组} }
	}
child{ node {齐次线性方程组} 
	child{ node {解空间} 
		child{ node {子空间}} 	
	}
	child{ node {基础解系} 
		child{ node {子空间的基}} 	
	}
}; 


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\end{document}
